832 research outputs found
Virtual Network Embedding Approximations: Leveraging Randomized Rounding
© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.The Virtual Network Embedding Problem (VNEP) captures the essence of many resource allocation problems. In the VNEP, customers request resources in the form of Virtual Networks. An embedding of a virtual network on a shared physical infrastructure is the joint mapping of (virtual) nodes to physical servers together with the mapping of (virtual) edges onto paths in the physical network connecting the respective servers. This work initiates the study of approximation algorithms for the VNEP for general request graphs. Concretely, we study the offline setting with admission control: given multiple requests, the task is to embed the most profitable subset while not exceeding resource capacities. Our approximation is based on the randomized rounding of Linear Programming (LP) solutions. Interestingly, we uncover that the standard LP formulation for the VNEP exhibits an inherent structural deficit when considering general virtual network topologies: its solutions cannot be decomposed into valid embeddings. In turn, focusing on the class of cactus request graphs, we devise a novel LP formulation, whose solutions can be decomposed. Proving performance guarantees of our rounding scheme, we obtain the first approximation algorithm for the VNEP in the resource augmentation model. We propose different types of rounding heuristics and evaluate their performance in an extensive computational study. Our results indicate that good solutions can be achieved even without resource augmentations. Specifically, heuristical rounding achieves 77.2% of the baseline’s profit on average while respecting capacities.BMBF, 01IS12056, Software Campus GrantEC/H2020/679158/EU/Resolving the Tussle in the Internet: Mapping, Architecture, and Policy Making/ResolutioNe
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
Algorithmic Aspects of Switch Cographs
This paper introduces the notion of involution module, the first
generalization of the modular decomposition of 2-structure which has a unique
linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm
and we take advantage of the involution modular decomposition tree to state
several algorithmic results. Cographs are the graphs that are totally
decomposable w.r.t modular decomposition. In a similar way, we introduce the
class of switch cographs, the class of graphs that are totally decomposable
w.r.t involution modular decomposition. This class generalizes the class of
cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We
use our new decomposition tool to design three practical algorithms for the
maximum cut, vertex cover and vertex separator problems. The complexity of
these problems was still unknown for this class of graphs. This paper also
improves the complexity of the maximum clique, the maximum independant set, the
chromatic number and the maximum clique cover problems by giving efficient
algorithms, thanks to the decomposition tree. Eventually, we show that this
class of graphs has Clique-Width at most 4 and that a Clique-Width expression
can be computed in linear time
Bicriteria Network Design Problems
We study a general class of bicriteria network design problems. A generic
problem in this class is as follows: Given an undirected graph and two
minimization objectives (under different cost functions), with a budget
specified on the first, find a <subgraph \from a given subgraph-class that
minimizes the second objective subject to the budget on the first. We consider
three different criteria - the total edge cost, the diameter and the maximum
degree of the network. Here, we present the first polynomial-time approximation
algorithms for a large class of bicriteria network design problems for the
above mentioned criteria. The following general types of results are presented.
First, we develop a framework for bicriteria problems and their
approximations. Second, when the two criteria are the same %(note that the cost
functions continue to be different) we present a ``black box'' parametric
search technique. This black box takes in as input an (approximation) algorithm
for the unicriterion situation and generates an approximation algorithm for the
bicriteria case with only a constant factor loss in the performance guarantee.
Third, when the two criteria are the diameter and the total edge costs we use a
cluster-based approach to devise a approximation algorithms --- the solutions
output violate both the criteria by a logarithmic factor. Finally, for the
class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms
for a number of bicriteria problems using dynamic programming. We show how
these pseudopolynomial-time algorithms can be converted to fully
polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur
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